OFFSET
0,2
COMMENTS
Binomial transform of A116383.
The substitution x-> x/(1+x+x^2) in the g.f. (this might be called an inverse Motzkin transform) yields the g.f. of A074331. - R. J. Mathar, Nov 10 2008
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n,j-k)*C(j,n-j).
Conjecture: n*(17*n-142)*a(n) + (17*n^2 + 95*n + 138)*a(n-1) + (-391*n^2 + 2488*n - 2908)*a(n-2) + (-17*n^2 - 603*n + 1892)*a(n-3) + 2*(697*n-2021)*(n-4)*a(n-4) + 60*(17*n-47)*(n-4)*a(n-5) = 0. - R. J. Mathar, Nov 15 2011
a(n) ~ (1+sqrt(5))^n * (5+sqrt(5)) / 10. - Vaclav Kotesovec, Feb 08 2014
MATHEMATICA
Table[Sum[Binomial[n, j-k]Binomial[j, n-j], {k, 0, n}, {j, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Feb 08 2012 *)
PROG
(PARI) {a(n) = sum(k=0, n, sum(j=0, n, binomial(n, j-k)*binomial(j, n-j)))}; \\ G. C. Greubel, May 23 2019
(Magma) [(&+[ (&+[Binomial(n, j-k)*Binomial(j, n-j): j in [0..n]]) : k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 23 2019
(Sage) [sum( sum(binomial(n, j-k)*binomial(j, n-j) for j in (0..n)) for k in (0..n)) for n in (0..30)] # G. C. Greubel, May 23 2019
(GAP) List([0..30], n-> Sum([0..n], k-> Sum([0..n], j-> Binomial(n, j-k)*Binomial(j, n-j) ))) # G. C. Greubel, May 23 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Feb 12 2006
STATUS
approved